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MATHEMATICS
-
I

MEAN VALUE THEOREMS
FUNCTIONS OF SINGLE

&

SEVERAL VARIABLES

I YEAR B.TECH








By

Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.







SYLLABUS OF MATHEMATICS
-
I

(
AS PER JNTU HYD
)

Name of the Unit

Name of the Topic

Unit
-
I

Sequences and Series


Basic definition of sequences and series


Convergence and divergence.


Ratio test


Comparison test


Integral test


Cauchy’s root test


Raabe’s test


Absolute and conditional convergence

Unit
-
II

Functions of single variable


Rolle’s theorem


Lagrange’s Mean value theorem


Cauchy’s Mean value theorem


Generalized mean value theorems


Functions of several variables


Functional dependence, Jacobian


Maxima and minima of function of two variables

Unit
-
III

Application of single variables


Radius , centre and Circle of curvature


Evolutes and Envelopes


Curve Tracing
-
Cartesian Co
-
ordinates


Curve Tracing
-
Polar Co
-
ordinates


Cu
rve Tracing
-
Parametric Curves

Unit
-
IV

Integration and its
applications


Riemann Sum


Integral representation for lengths


Integral representation for Areas


Integral representation for Volumes


Surface areas in Cartesian and Polar co
-
ordinates


Multiple
integrals
-
double and triple


Change of order of integration


Change of variable

Unit
-
V

Differential equations of first
order and their applications


Overview of differential equations


Exact and non exact differential equations


Linear differential equations


Bernoulli D.E


Newton’s Law of cooling


Law of Natural growth and decay


Orthogonal trajectories and applications

Unit
-
VI

Higher order Linear D.E and
their applications


Linear D.E of second and higher order with constant coefficients


R.H.S term of the form
e
xp(ax)


R.H.S term of the form
sin ax and cos ax


R.H.S term of the form exp(ax) v(x)


R.H.S term of the form exp(ax) v(x)


Method of variation of parameters


Applications on bending of beams, Electrical circuits and simple harmonic motion

Unit
-
VII

Laplace
Transformations


LT of standard functions


Inverse LT

first shifting property


Transformations of derivatives and integrals


Unit step function, Second shifting theorem


Convolution theorem
-
periodic function


Differentiation and integration of transforms


Applic
ation of laplace transforms to ODE

Unit
-
VIII

Vector Calculus


Gradient, Divergence, curl


Laplacian and second order operators


Line, surface , volume integrals


Green’s Theorem and applications


Gauss Divergence Theorem and applications


Stoke’s Theorem and
applications




CONTENTS

UNIT
-
2

Functions of Single & Several Variables




Rolle’s Theorem(without Proof)



Lagrange’s Mean Value Theorem(without Proof)



Cauchy’s Mean Value Theorem(without Proof)



Generalized Mean Value Theorem (without Proof)



Functions of
Several Variables
-
Functional dependence



Jacobian



Maxima and Minima of functions of two variables





Introduction

Real valued function
: Any function

is called a Real valued function.




Limit of a function:

Let

is a real valued function and
. Then, a real number

is
said to be limit of

at

if for each

whenever
. It is denoted by
.

Continuity:
Let

is a real valued function and
. Then,

is said to be continuous at

if for each

whenever
. It is
denoted by
.

Note:

1) The function
is continuous every where

2) Every

function is continuous every where.

3) Every

function is not continuous, but in

the function

is continuous.

4) Every polynomial is continuous every where.

5) Every exponential function is continuous every where.

6) Every lo
g function is continuous every where.

Differentiability:

Let

be a function, then

is said to be differentiable or derivable at
a point
, if

exists.






Functions
Algebraic
Trignonometric
Inverse
Exponential
Logarithmic
Hyperboli
c


MEAN VALUE THEOREMS

Let

be a given function.

ROLLE’S THEOREM

Let

such that

(i)


is continuous in

(ii)


is differentiable (or) derivable in

(iii)


then

atleast one point

in

such that

Geometrical Interpretation of Rolle’s Theorem:

From the diagram, it can be observed that

(i) there is no gap for the curve

from
. Therefore, the function is
continuous

(ii) There exists unique tangent for every intermediate point between

and

(ii) Also the ordinates of

and

are same, then by Rolle’s theorem, there exists atleast one point

in between
and

such that tangent at

is parallel to
axis.







LAGRANGE’S MEAN VALUE THEOREM

Let

such that

(i)


is continuous in

(ii)


is differentiable (or) derivable in

then

atleast one point


such that














Geometrical Interpretation of Rolle’s Theorem:

From
the diagram, it can be observed that

(i) there is no gap for the curve

from
. Therefore, the function is
continuous

(ii) There exists unique tangent for every intermediate point between

and

T
hen by
Lagrange
’s

mean value

theorem, there exists atleast one point

in between
and

such that tangent at

is parallel to
a straight line joining the points

and








CAUCHY’S
MEAN VALUE THEOREM

Let

are such that

(i)


are

continuous in

(ii)


are

differentiable (or) derivable in

(iii)


then

atleast one point


such that

Generalised
Mean Value Theorems

Taylor’s Theorem:

If
such that

(i)

are continuous on

(ii)

are derivable or differentiable on

(iii)

, then

such that


where,

is called as Schlomilch


Roche’s form of remainder and is given by















Lagrange’s form of Remainder:

Substituting

in

we get Lagrange’s form of Remainder

i.e.




Cauchy’s form of Remainder:

Substituting

in

we get Cauchy’s form of Remainder

i.e.


Maclaurin’s Theorem:

If
such that

(i)

are continuous on

(ii)

are derivable or differentiable on

(iii)

, then

such that


where,

is called as Schlomilch


Roche’s form of remainder and is given by




Lagrange’s form of Remainder:

Substituting

in

we get Lagrange’s form of Remainder

i.e.




Cauchy’s form of Remainder:

Substituting

in

we get Cauchy’s form of Remainder

i.e.













Jacobian

Let

are two functions , then the Jacobian of

w.r.t

is
denoted by

and is defined as



Properties:






If

are functions of

and

are functions of

then



Functional Dependence:

Two functions

are said to be functional dependent on one another if the Jacobian of

is zero.

If they are functionally dependent on one another, then it is possible to find the relation between
these two functions.

MAXIMA AND MINIMA

Maxima and Minim
a for the function of one Variable:

Let us consider a function

To find the Maxima and Minima, the following procedure must be followed:

Step 1:

First find the first derivative and equate to zero. i.e.

Step 2:

Since

is a polynomial

is a polynomial equation. By solving this
equation we get roots.

Step 3:

Find second derivative i.e.

Step 4:

Now substitute the obtained roots in

Step 5:

Depending on the Nature of

at that point we will solve further. The following cases will
be there.





Case (i):

If

at a point say

, then

has maximum at

and the maximum
value is given by

Case (ii):

If

at a point say

, then

has minimum at

and the minimum
value is given by

Case (iii):

If

at a point say

, then

has neither minimum nor maximum. i.e.
stationary.


Maxima and Minima for the function of Two Variable

Let us consider a funct
ion

To find the Maxima and Minima for the given function, the following procedure must be followed:

Step 1:

First find the first derivatives and equate to zero. i.e.

(
Here, since we have two variables, we go fo
r partial derivatives, but not ordinary
derivatives
)

Step 2:

By solving



, we get the different values of

and
.

Write these values as set of ordered pairs. i.e.

Step 3:

Now, find second order partial derivatives.

i.e.,


Step 4:

Let us consider

,

Step 5:

Now, we have to see for what values of
, the given function is maximum/minimum/


does not have extreme values/
fails to have maximum or minimum.



If at a point, say
, then

has
maximum

at this point and the
maximum value will be obtained by substituting

in the given function.



If at a point, say
, then

has
minimum

at this point and the
m
in
imum value will be obtained by substituting

in the given function.



If at a point, say
, then

has

neither maximum nor minimum

and
such points are called as saddle points
.



If at a p
oint, say
, then

fails to have
maximum
or minimum and case
needs further investigation to decide maxima/minima
.

i.e.

No conclusion





Problem

1)

Examine the function for extreme values


Sol:
Given

The first order partial derivatives of

are given by


and



Now, equating first order partial derivatives to zero, we get





Solving

we get



Substituting

in



Substituting

in



All possible set of values are

Now, the second order partial derivatives are given by







Now,

At a point

&


has maximum at

and the maximum value will be obtained by substituting

in the
function


i.e.

is the maximum value.




Also, at a point

&


has minimum at

and the minimum value is
.

Al
so, at a point



has neither minimum nor maximum at this point.

Again, at a point



has neither minimum nor maximum at this point.


Lagrange’s Method of Undetermined Multipliers

This method is useful to find the
extreme values (i.e., maximum and minimum) for the given
function, whenever some condition is given involving the variables.

To find the Maxima and Minima for the given function using Lagrange’s Method , the following
procedure must be followed:

Step 1:

Le
t us consider given function to be

subject to the condition

Step 2:

Let us define a Lagrangean function

, where

is called the Lagrange
multiplier.

Step 3:

Find first order partial derivatives and equate to zero


i.e.







Let the given condition be

Step 4
:

Solve

, eliminate

to get the values of

Step 5:

The values so obtained will give the stationary point of

Step 6:

The minimum/maximum value will be obtained by substituting the values of

in the
given function.

Problem

1) Find the minimum value of

su
bject to the condition

Sol:

Let us consider given function to be

and

Let us define Lagrangean function

, where

is called the Lagrange multiplier.






Now,







Solving


Now, consider


Again, consider


Again solving




Given






Similarly, we get

Hence, the minimum
value of the function is given by